(SEM V) THEORY EXAMINATION 2022-23 DIGITAL SIGNAL PROCESSING

Course: B.Tech (Semester V)                                                     Subject Code: KEE-057

Subject Name: Digital Signal Processing                                 Time: 3 Hours

Total Marks: 100

Note: Attempt all sections. If any data is missing, assume suitably. 

Section A – Short Answer Questions (2 × 10 = 20 Marks)

Answer all briefly:

Find the linear convolution of x1(n)={1,2,3,4}x_1(n) = \{1,2,3,4\}x1​(n)={1,2,3,4} and x2(n)={1,1,2,2}x_2(n) = \{1,1,2,2\}x2​(n)={1,1,2,2}.

For x(n)={4,−2,4,−6}x(n) = \{4, -2, 4, -6\}x(n)={4,−2,4,−6}, sketch its odd and even parts for −2≤n≤1-2 \le n \le 1−2≤n≤1.

State the Nyquist Sampling Theorem.

Explain, with block diagram, the process of Analog-to-Digital Conversion.

Define properties of convolution in an LTI system.

Define the Twiddle Factor and list two of its properties.

Differentiate between FIR and IIR filters with examples.

Define Frequency Warping in the Bilinear Transformation method.

Illustrate the symmetry and periodicity properties of phase factor WNW_NWN​ used in FFT.

Compute the DFT of x(n)=cos⁡(nπ/2)x(n) = \cos(n\pi/2)x(n)=cos(nπ/2) for N=4N = 4N=4 using the DIF-FFT algorithm. 

Section B – Descriptive Questions (10 × 3 = 30 Marks)

Attempt any three:

(i) Check whether the system y(n)=ex(n)y(n) = e^{x(n)}y(n)=ex(n) is linear/nonlinear, static/dynamic, and time invariant/variant.
(ii) Check the stability of filter

1. H(z)=1+2z−11−0.5z−1+0.2z−2H(z) = \frac{1 + 2z^{-1}}{1 - 0.5z^{-1} + 0.2z^{-2}}H(z)=1−0.5z−1+0.2z−21+2z−1​

Explain discrete-time processing of a continuous-time signal with the help of a block diagram.

Determine the impulse response of the system given by

1. y(n)=x(n)+3x(n−1)−4x(n−2)+2x(n−3)y(n) = x(n) + 3x(n-1) - 4x(n-2) + 2x(n-3)y(n)=x(n)+3x(n−1)−4x(n−2)+2x(n−3)

Design an IIR filter using the Impulse Invariant Method for

1. H(s)=9(s+0.2)s2+0.2s+2H(s) = \frac{9(s + 0.2)}{s^2 + 0.2s + 2}H(s)=s2+0.2s+29(s+0.2)​

with T=1T = 1T=1 sec.

Differentiate between Wavelet Transform and Fourier Transform and list applications of Wavelet Cosine Transform. 

 Section C – Long Answer Questions (10 × 5 = 50 Marks)

Q3

(a) For an LTI system with h(n)=anh(n) = a^nh(n)=an for n≥0n \ge 0n≥0, find the response to input x(n)=u(n)−u(n−N)x(n) = u(n) - u(n-N)x(n)=u(n)−u(n−N).
(ii) Check whether the system F[x(n)]=ax(n)+bx2(n)F[x(n)] = ax(n) + bx^2(n)F[x(n)]=ax(n)+bx2(n) is linear and time invariant.
or
(b) Explain any two IIR filter realization methods with suitable examples.

Q4

(a) Derive the Sampling Theorem and explain Aliasing.
or
(b) Explain Multirate Signal Processing in detail.

Q5

(a) Compute the Circular Convolution of
x1(n)=[1,2,3,4]x_1(n) = [1, 2, 3, 4]x1​(n)=[1,2,3,4] and x2(n)=[1,1,2,2]x_2(n) = [1, 1, 2, 2]x2​(n)=[1,1,2,2]
using the graphical method, and verify with DFT and IDFT.
or
(b) Determine magnitude and phase responses of the system

y(n)+y(n−1)=x(n)−2x(n−1)y(n) + y(n-1) = x(n) - 2x(n-1)y(n)+y(n−1)=x(n)−2x(n−1) 

Q6

(a) Design a Low Pass Filter with the desired frequency response

Hd(ejω)={e−j4ω,∣ω∣≤π/40,otherwiseH_d(e^{j\omega}) = \begin{cases} e^{-j4\omega}, & |\omega| \le \pi/4 \\ 0, & \text{otherwise} \end{cases}Hd​(ejω)={e−j4ω,0,​∣ω∣≤π/4otherwise​

using the Rectangular Window w(n)=1w(n) = 1w(n)=1 for 0≤n≤40 \le n \le 40≤n≤4.
Find coefficients hd(n)h_d(n)hd​(n) and frequency response H(ejω)H(e^{j\omega})H(ejω).
or
(b) Determine H(z)H(z)H(z) for a Butterworth filter satisfying:

0.707≤∣H(ejω)∣≤1,0≤ω≤π/20.707 \le |H(e^{j\omega})| \le 1, \quad 0 \le \omega \le \pi/20.707≤∣H(ejω)∣≤1,0≤ω≤π/2 ∣H(ejω)∣≤0.2,3π/4≤ω≤π|H(e^{j\omega})| \le 0.2, \quad 3\pi/4 \le \omega \le \pi∣H(ejω)∣≤0.2,3π/4≤ω≤π

Use the Impulse Invariant Transformation method with T=1T = 1T=1 sec.

Q7

(a) Draw the flow graph for 8-point DIT-FFT of sequence
x(n)={1,2,3,4,4,3,2,1}x(n) = \{1, 2, 3, 4, 4, 3, 2, 1\}x(n)={1,2,3,4,4,3,2,1}.
or
(b) Explain Radix-2 DIT-FFT Algorithm and compare it with DIF-FFT.